While both convergence rates are concerned with large sample properties of a statistical procedure, they are two distinct subjects. It is widely perceived that posterior contraction rates appear to be a stronger result than convergence rates when \(\hat{\theta }_n\)  is a Bayes estimator, a point estimator that minimizes the posterior expected loss. Indeed, convergence of the posterior at a rate ensures the existence of point estimators that converge to the true \(\theta _0\)  at the same rate. For example, if \(\epsilon _n\)  is a contraction rate, the center \(\hat{\theta }_n\)  of the smallest ball that contains posterior mass at least \(1/2\)  may achieve the same convergence rate (cf. Theorem 2.5 in [1]}). In addition, if we consider a bounded metric \(d\)  whose square is convex, the posterior mean \(\hat{\theta }_n:=\int \theta d\Pi _n(\theta \mid X^{(n)})\) , which is a special case of Bayes estimators under the squared error loss [2]}, satisfies that
\(d(\hat{\theta }_n,\theta _0)\le M_n\epsilon _n+\Vert d\Vert _\infty ^{1/2}\Pi _n(\theta :d(\theta ,\theta _0)>M_n\epsilon _n\mid \mathbb {X}^{(n)}),\) 
